Question

Maximum value of array factor for N-element linear array occurs at ______

(a) \(θ_m=cos^{-1}⁡(\frac{λ}{2πd}[-β±2πm])\)

(b) \(θ_m=cos^{-1}⁡(\frac{λ}{2πd}[β±2πm])\)

(c) \(θ_m=sin^{-1}⁡(\frac{λ}{2πd}[-β±2πm])\)

(d) \(θ_m=sin^{-1}(\frac{λ}{2πd}[-β±2πm])\)

The question was asked in an international level competition.

Origin of the question is N-element Linear Array in section Antenna Array of Antennas

Answer 1

The correct choice is (a) \(θ_m=cos^{-1}⁡(\frac{λ}{2πd}[-β±2πm])\)

Explanation: Normalized array factor is given by \(\frac{sin(Nᴪ/2)}{Nsin(\frac{ᴪ}{2})}.\)

To get maximum denominator is equal to zero.

⇨ \(sin(\frac{ᴪ}{2})=0  \)

⇨ \(\frac{ᴪ}{2}=±nπ \)

⇨ kdcosθ+β=±2πm

⇨ \(θ_m=cos^{-1}⁡(\frac{λ}{2πd}[-β±2πm]).\)